Perishable Inventory System with Server Interruptions, Multiple Server Vacations, and N Policy

Perishable Inventory System with Server Interruptions, Multiple Server Vacations, and N Policy

K. Jeganathan (University of Madras, Chennai, India), N. Anbazhagan (Department of Mathematics, Alagappa University, Karaikudi, India) and B. Vigneshwaran (Department of Mathematics, Thiagarajar College of Engineering, Madurai, India)
DOI: 10.4018/ijoris.2015040103

Abstract

This article presents a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size . The authors assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. It is assumed that demand for the commodity is of unit size. The service starts only when the customer level reaches a prefixed level , starting from the epoch at which no customer is left behind in the system. The arrivals of customers to the service station form a Poisson process. The server goes for a vacation of an exponentially distributed duration whenever the waiting area is zero. If the server finds the customer level is less than when he returns to the system, he immediately takes another vacation. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The service process is subject to interruptions, which occurs according to a Poisson process. The interrupted server is repaired at an exponential rate. Also the waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the number customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived and the total expected cost is also considered. The results are illustrated with numerical examples.
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Introduction

In most of the inventory models considered in the literature, the demanded items are directly issued from the stock, if available. The demands that occurred during stock out period are either not satisfied (lost sales case) or satisfied only after the receipt of the ordered items (backlog case). In the latter, it is assumed either all (full backlog case) or only a prefixed number of demands (partial backlogging) that occurred during stock out period are satisfied. For review of these works see (Nahmias, 1982), (Raafat, 1991), (Kalpakam & Arivarignan, 1990), (Elango & Arivarignan, 2003), (Liu & Yang, 1999), (Cakanyildirim et al., 2000), (Goyal & Giri, 2001) and (Dura et al., 2004), and the references therein. But in the case of inventories maintained at service facilities, the demanded items are issued to the customers only after some service is performed on it. In this situation the items are issued not at the time of demand but after a random time of service. This forces the formation of queues in these models. This necessitates the study of both the inventory level and the queue length joint distributions. Study of such models is beneficial to organizations which:

  • Provide service to customers by using items from a stock;

  • Maintain stock of items each of which needs service such as assembly or initialization or installation, etc.

Examples for the first type include firms that are engaged in servicing consumer products such as Television sets, Computers, etc., and for the second type include firms that supply bicycles which need assembly of its parts, that supply food items which need heating or garnishing and that computers which need installation of basic services.

(Krishnamoorthy & Anbazhagan, 2008), analyzed a perishable queueing-inventory system with policy, Poisson arrivals, exponential distributed lead times and service times. The joint probability distributions of the number of customers in the system and the inventory level were obtained in the steady state case. Two other papers where an inventory model with service time is considered, are by (Krishnamoorthy et al., 2010, 2012). The article (Krishnamoorthy et al., 2010), is considered for an inventory model with instantaneous replenishment and the service process is subject to interruptions. The discussion in (Krishnamoorthy et.al., 2012), is an inventory model with positive lead time, server interruptions and an orbit of infinite capacity, where no waiting space is provided for customers, other than for the one whose service gets interrupted.

In many real world queueing systems, server(s) may become unavailable for a random period of time when there are no customers in the waiting line at a service completion instant. This random period of server absence, often called a server vacation can represent the time of server’s performing some secondary task. This has been extensively investigated (Tian & Zhang, 2006), (Takagi, 1991, 1993) and (Doshi, 1986, 1990). (Daniel & Ramanarayanan, 1987), have first introduced the concept of server vacation in inventory with two servers. In (Daniel & Ramanarayanan, 1988), they have studied an inventory system in which the server takes a rest when the level of the inventory is zero. (Sivakumar, 2011), analyzed a retrial inventory system with multiple server vacations. Recently, (Narayanan et al., 2008), considered an inventory system with random positive service time. Customers arrived to the service station according to a Markovian arrival process and service times for each customers had phase-type distribution.

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